Question

Let $R$ and $S$ be any two equivalence relations on a set $X$ . Then which of the following is incorrect statement

• A. $R\cup S$ is an equivalence relation on $X$
• B. $R^{-1}$ is an equivalence relation on $X$
• C. $R^{-1}\cap S^{-1}$ is an equivalence relation on $X$
• D. $\Delta$ is an equivalence relation on $X$ , where $\Delta$ is the diagonal relation on $X$ .

Answer 1

Answer: (A)

We know that the union of two equivalence relation on a set is not necessarily an equivalence relation on the set.
For example let $x=\left\{a,b,c\right\}$
$R=\left\{\left(a,a\right),\left(b,b\right)\left(c,c\right),\left(a,b\right),\left(b,a\right)\right\}$
$S=\left\{\left(a,a\right),\left(b,b\right)\left(c,c\right),\left(b,c\right),\left(c,b\right)\right\}$
Clearly, $R$ and $S$ are equivalence relation
But $R\cup S$ is not transitive because $\left(a,b\right)\in R\cup S$
and $\left(b,c\right)\in R\cup S$ but $\left(a,c\right)\notin R\cup S$
Hence, $R\cup S$ is not equivalence relation